The QP in (REF ), which involves minimizing a quadratic over a sphere, is called a trust region subproblem (TRS) [1]}. It is well-known from the TRS literature (see, e.g., Theorem 1.1
in [2]}) that \(\mathbf {u}\)  is a
solution to the TRS in (REF ) if and only if there exists \(\lambda ^\ast \ge 0\)  such that
\(| \mathbf {u}^\ast | \le \Omega _{\text{SO}},\\\big (\mathbf {A}_{\mu } + \lambda ^\ast \mathbf {I}\big ) \mathbf {u}^\ast = -\mathbf {G}_{\mu }(\mathbf {p},s),\\\lambda ^\ast (\Omega _{\text{SO}} - | \mathbf {u}^\ast |) = 0,\\\mathbf {A}_{\mu } + \lambda ^\ast \mathbf {I} \succeq \mathbf {0}.\) 
